In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex. It has Schläfli symbol of t1{6,3}; its edges form an infinite arrangement of lines.[1][2] It can also be constructed as a cantic hexagonal tiling, h2{6,3}, if drawn by alternating two colors of triangles.

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Kagome (籠目) is a traditional Japanese woven bamboo pattern; its name is composed from the words kago, meaning "basket", and me, meaning "eye(s)", referring to the pattern of holes in a woven basket. A kagome lattice is an arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. Despite the name, these crossing points do not form a mathematical lattice.

Some minerals, namely jarosites and herbertsmithite, contain layers with kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism. The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice in two or three dimensions. The term "kagome lattice" in this context was coined by Japanese physicist Kōji Fushimi, who was working with Ichirō Shōji. The first paper[4] on the subject appeared in 1951.[5]

A tiling with alternate large and small triangles is topologically identical to the trihexagonal tiling, but has a different symmetry group. The hexagons are distorted so 3 vertices are on the mid-edge of the larger triangles. As with the trihexagonal tiling, it has two uniform colorings. The hexagons can also be distorted into triangular star polygons as shown in the third example.

The trihexagonal tiling is also one of eight uniform tilings that can be formed from the regular hexagonal tiling (or the dual triangular tiling) by a Wythoff construction. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

The rhombille tiling,[6] also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles. It is the dual tiling of the trihexagonal tiling.